Theorem univ 4596

Description: The union of the universe is the universe. Exercise 4.12(c) of [Mendelson] p. 235. (Contributed by NM, 14-Sep-2003.)

Assertion

Ref	Expression
univ	⊢ ∪ V = V

Proof of Theorem univ

Step	Hyp	Ref	Expression
1		pwv 3912	. . 3 ⊢ 𝒫 V = V
2	1	unieqi 3923	. 2 ⊢ ∪ 𝒫 V = ∪ V
3		unipw 4332	. 2 ⊢ ∪ 𝒫 V = V
4	2, 3	eqtr3i 2255	1 ⊢ ∪ V = V

Syntax hints:   = wceq 1398  Vcvv 2812  𝒫 cpw 3668  ∪ cuni 3913

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4227  ax-pow 4286

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-rex 2526  df-v 2814  df-in 3216  df-ss 3223  df-pw 3670  df-sn 3694  df-uni 3914

This theorem is referenced by: (None)